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# Class Note for EMSE 388 at GW

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This 25 page Class Notes was uploaded by an elite notetaker on Saturday February 7, 2015. The Class Notes belongs to a course at George Washington University taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/07/15

EMSE 388 Quantitative Methods in Cost Engineering Chapter 3 FITTING EXPONENTIAL GROWTH Often a SCATTERPLOT will indicate that x and y are related but not in a linear fashion EXAMPLEMCROSOFT SALES Not Real Data 4500 Microsoft Sales 4000 3500 3000 2500 2000 1500 1000 500 Salesin millions 12 Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 3 Page 81 EMSE 388 Quantitative Methods in Cost Engineering When slope of the curve is increasing it may be reasonable to fit an Exponential Growth Curve b Yzae x where ab are constant parameters Microsoft Sales 3500 H y 64486604252X R2 09536 Salesin millions N O O O l Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 82 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Examples Exponential Curves Positive Exponent Y Increases when X Increases Y2e3x 35000 30000 25000 20000 15000 10000 5000 0 05 1 15 2 25 3 35 Y2e3x Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 83 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Negative Exponent Y Decreases when X Increases Y2EXP15Q 25 15 1 05 0 05 1 15 2 25 3 Y2EXP15X Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 84 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering IMPORTANT PROPERTY OF EXPONENTIAL GROWTH 7 Prediction forY at x YOC 616 x 7 b 7 7 Prediction forY at x1 Y x 1 2 ae 1 6 ae x 6 Y x Results in ABSOLUTE INCREASE Of 11xDekgt1 bgt0 Yx lt1 9 lt 0 RESULTS in a PERCENTAGE INCREASE Of Yx1 Yx 1 Yo gt0 bgt0 lt0 blt0 CONCLUSION Absolute Increase and Percentage Increase in Y are constant per unit increase in x regardless of the current value of x MEMORYLESS PROPERTY Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 85 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Example Microsoft Sales Microsoft Exponential Trend in millions of Microsoft Sales in Exponential Year Observation Millions of Growth Check 1995 1 80 1996 2 120 1 50 1997 3 224 1 87 1998 4 434 1 94 1999 5 739 1 70 2000 6 751 1 02 2001 7 1367 1 82 2002 8 1781 1 30 2003 9 2759 1 55 2004 10 3876 1 40 Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 86 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Fitting Exponential Growth Curve Results in Yx 61598 604285 Microsoft Sales h h O 01 O O O O l l l l y 61 598e04285X R2 09818 A A O 01 O O O O l l l l Salesin millions l O O O Year What is the interpretation of R2 in this case Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 87 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering EXPLANATION of R2 in Exponential Growth Y aem ltgtLnY Lna 9 x CONCLUSION Sales Y follows an Exponential Growth Curve if and only if the natural logarithm of Sales LnY follows a straight line Note that b equals the SLOPE of the linear line and Lna is the INTERCEPT of the linear line This can be easily verified using the SLOPE and INTERCEPT function in Excel Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 88 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering RETURNING TO MICROSOFT SALES EXAMPLE Microsoft REGRESSION Sales in FlT LnSalesLnSales Year Observation Millions of LNSALES LNSALES Average Fit2 1995 1 80 4382 4549 4389 0028 1996 2 120 4787 4978 2855 0036 1997 3 224 5412 5406 1135 0000 1998 4 434 6073 5834 0163 0057 1999 5 739 6605 6263 0016 0117 2000 6 751 6621 6691 0021 0005 2001 7 1367 7220 7120 0552 0010 2002 8 1781 7485 7548 1016 0004 2003 9 2759 7923 7977 2089 0003 2004 10 3876 8263 8405 3188 0020 Mean LnSales 6477 15425 0281 SS SSE 2 SS SSE Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 89 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 9000 8000 7000 6000 5000 4000 3000 2000 1000 0000 Natural Logarithm of Microsoft Sales y 04285x 41206 R2 09818 12 Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 3 Page 90 EMSE 388 Quantitative Methods in Cost Engineering MAKING PREDICTIONS WITH THE EXPONENTIAL GROWTH LINE What would be an estimate for the sales of Microsoft in 2004 Year 11 Predicted Microsoft Sales 61 5982Exp04285Year 61 5982Exp042851 1 686127 Prediction may also be calculated using the GROWTH function in Excel CAN WE USE THIS REGRESSION TO PREDICT MANY YEARS AHEAD WHAT ABOUT THE ACCURACY OF THE PREDICTION GIVEN THE POINT ESTIMATES OF REGRESSION PARAMETES Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 91 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering OR BETTER WHAT ABOUT THE UNCERTAINTY OF THE PREDICTION GIVEN THE POINT ESTIMATES OF REGRESSION PARAMETES MAKE NORMAL ASSUMPTION OF RESIDUALS STEP 5A A LnYELnYamp b XgampX13g Model the Error Term Residuals as normal distributed random variable with mean u0 and standard deviation s8 such that Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 92 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering COINCIDES WITH METHOD OF MOMENTS FOR ESTIMATING DISTRIBUTIONAL PARAMETERS STANDARDIZED RESIDUALSRESDUAL2 RESIDUALS 016706 002791 089218 019005 003612 101496 000565 000003 003015 023859 005692 127416 034239 011723 182848 006996 000489 037363 010055 001011 053698 006335 000401 033831 00541 1 000293 028898 014263 002034 076172 000000 003506 018725 MEAN RESIDUALS se2 se Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 93 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering STEP 6A Shift normal distribution using predicted value Hence in 2004 Year 11 the natural logarithm of predicted sales using the least squares fit LNY04285X41206 8 042851141206 8 88336 8 CONCLUSION LnY at 12 years is Normal Distributed with Mean 88336 and variance s2e 003506 Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 94 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Distribution of Prediction LnSales at 2004 11 years 10000 95 8000 7 777777777777777777777777777777777777777777777777777777777777777777777777 7 3 VI 7 6000 7 777777777777777777777777777777777777777777777777777777777777777777777777 7 0 g 50 I E 4000 7 7777777777777777777777777777 7139 77777777777777777777777777 7 77777777777777 7 V 39 2000 7 77777777777777777777777777 7777777777777777777777777 77777777777777 7 5I E i 000 85000E 86000 87000 88000 89000 90000 91000 92000 93000 39 I 5 MW 85256 88336 91415 Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 95 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering We read from the graph or using the NORMINV function in EXCEL that o PrLnSales g 8336 050 o 90 Credibility Interval for LNSales is 8525691416 HOWEVER We are not interested in LnSales but we are interested in Sales Realizing that PrLnY Sx pltgt PrY S ex 2p PrlSLnYSupltgtPrelSYSe p We calculate that o PrSales g Exp8336 PrSales 3 686127 050 90 Credibility Interval for Sales is Exp85256Exp91416 504245933614 Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 96 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering LnSales Sales Lower Credibility Limit 005 85256 5042447 Upper Credibility Limit 095 91416 9336144 ARE WE SATISFIED WITH THE ANALYSIS What about the normality assumption of the residuals CREATING A NORMAL PROBABILITY PLOT 1 Normal Distribution Assumption of Residuals Residuals are normal distributed random variable with mean u0 and standard deviation s8 Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 97 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 2 Rescale Residuals to Standardized Residuals by dividing by ss Hence with the assumption above Standardized Residuals are Standard Normal Distributed with mean 0 and standard deviation 1 Le NO1 3 Order the Standardized Residuals such that 504 1 gm 8 8 lt lt lt S S S S 8 8 8 8 4 calculate 8 8 8n QUZF jjngF 7 gltngtF 0 S8 S8 S8 where F is the standard normal cumulative distribution function 5 Plot the points 1 1 2 1 n l l n l n a 1 n 2412 n 7 n 17 H 2401 Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 98 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering If Standardized Residuals are in fact N01 distributed these points should all be one the line yx Ordered Standard Rank Standardized Emperical CDF Normal CDF Observation Residuals Value Value 1 10150 500 1551 2 08922 1500 1861 3 07617 2500 2231 4 03736 3500 3543 5 03383 4500 3676 6 02890 5500 3863 7 00302 6500 5120 8 05370 7500 7044 9 12742 8500 8987 10 18285 9500 9663 Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 3 Page 99 EMSE 388 Quantitative Methods in Cost Engineering Normal Probability Plot 10000 8000 6000 4000 2000 Standard Normal CDF 000 000 2000 4000 6000 8000 10000 EM PERICAL CDF CONCLUSION Residuals do not appear to be normally distributed Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 100 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering ALTERNATIVE APPROACH TO ESTABLISH UNCERTAINTY PREDICTION 1 Use Empirical Distribution of Error Term Residuals 2 Shift Distribution of Residuals using predicted value of LNSales Empirical Distribution of LnSaes at 11 years 010 1 quot71 09 7 7777777777777 777777777777 7777777777777 7777777 77 08 07 7 7777777777777 7777777777 77 1 7777777777777 00 7 77777777777777 7777777777777 00 7 77737 777777777 7777777777777 04 7 7777777777777 77777777777 7777777777777 7777777777777 77 03 7 7777777777777 7777777777 7777777777777 02 7 7777777777777 77 77777777777 7777777777777 damph PrLnSales gy 4h 01 5 0 85000 87000 89000 91000 y 50 m o o o 86436 87749 91760 Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 101 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering From the graph we conclude that o PrLnSales 3 87749 050 o 90 Credibility Interval for LnSaIes is 8643491760 HOWEVER We are not interested in LnSaIes but we are interested in Sales Realizing that PrLnY Ex 19 ltgtPrY Sex 19 PrlSLnYSupltgtPrel SYSe p We calculate that o PrSales g Exp87749 PrSales 3 646991 050 90 Credibility Interval for Sales is Exp86434Exp91760 567369966275 Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 102 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering LnSaes Sales Lower Credibility Limit 005 86436 567369 Upper Credibility Limit 095 91760 966275 SUMMARY 0 Initial XY Scatter plot shows a dependence between X and Y but a non linear one Using a transformation in this case LNY we were able to transform to a scatter plot of LNY and X that looks linear 0 We used the tool box of linear regression analysis to fit a straight line to LNY 0 We are able to predict Y and uncertainty of this prediction using the inverse transformation Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 3 Page 103 EMSE 388 Quantitative Methods in Cost Engineering OBSERVATION This is a general approach to curve fitting One could besides transforming Y values also transform Xvalues Using a TrialError approach one creates a linear looking scatter plot and use linear regression afterwards Example Chapter 4 The PowerModel Fitting Model Y aX ltgt LnY Lna b LnX Hence 1 Transform Yvalues to LnY 2 Transform Xvalues to LnX 3 Scatter Plot of LnXLnY displays linear behavior Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 104 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Chapter 5 The Pearl Curve L is known upper Limit L Y Y CgtLn Ln a bX 1ae bX L Y Hence 1 Transform Yvalues to LnYLY 2 Scatter Plot of X LnYLY displays linear behavior Lecture Notes by Dr J Rene van Dorp Chapter 3 Page 105 Source Financial Models Using Simulation and Optimization by Wayne Winston

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